Problem 12
Question
For exercises 1-66, simplify. $$ \frac{3 x-12}{15 x} $$
Step-by-Step Solution
Verified Answer
The simplified form is \(\frac{x-4}{5x}\).
1Step 1: Identify the Numerator and Denominator
The given expression is \(\frac{3x - 12}{15x}\). Identify the numerator as \(3x - 12\) and the denominator as \(15x\).
2Step 2: Factor the Numerator
Look for common factors in the numerator. The numerator \(3x - 12\) can be factored as \(3(x - 4)\).
3Step 3: Rewrite the Expression
Substitute the factored form of the numerator back into the original expression: \(\frac{3(x-4)}{15x}\).
4Step 4: Simplify the Fraction
Factor out the common constants. Both the numerator and the denominator have a common factor of 3: \(\frac{3(x-4)}{15x} = \frac{x-4}{5x}\).
Key Concepts
algebraic expressionsfactoringnumerator and denominator
algebraic expressions
In algebra, we work with expressions that contain variables, numbers, and arithmetic operations. An algebraic expression is a mathematical phrase that can involve adding, subtracting, multiplying, and dividing.
Algebraic expressions can range from simple, like \(3x + 2\), to complex ones, like \(4x^2 - 7x + 5\). In the given exercise, \(3x - 12\) and \(15x\) are algebraic expressions. The goal is to simplify these expressions to their simplest form.
To simplify, it often helps to combine like terms and perform operations indicated by the expression. This process makes it easier to solve equations or other mathematical problems.
Overall, understanding how to manipulate algebraic expressions is a key skill in algebra.
Algebraic expressions can range from simple, like \(3x + 2\), to complex ones, like \(4x^2 - 7x + 5\). In the given exercise, \(3x - 12\) and \(15x\) are algebraic expressions. The goal is to simplify these expressions to their simplest form.
To simplify, it often helps to combine like terms and perform operations indicated by the expression. This process makes it easier to solve equations or other mathematical problems.
Overall, understanding how to manipulate algebraic expressions is a key skill in algebra.
factoring
Factoring involves breaking down an algebraic expression into simpler pieces that can be multiplied together to give the original expression. This is particularly useful for simplifying fractions.
In our example, the numerator \(3x - 12\) can be factored by looking for a common factor. Both terms \(3x\) and \(-12\) have a common factor of 3. We can then write the numerator as \(3(x - 4)\).
Factoring makes simplifying fractions easier because it helps to reveal common factors between the numerator and the denominator. By removing these common factors, we reduce the expression to its simplest form.
Mastering factoring is vital since it is widely used in solving quadratic equations, integrating functions, and simplifying algebraic fractions.
In our example, the numerator \(3x - 12\) can be factored by looking for a common factor. Both terms \(3x\) and \(-12\) have a common factor of 3. We can then write the numerator as \(3(x - 4)\).
Factoring makes simplifying fractions easier because it helps to reveal common factors between the numerator and the denominator. By removing these common factors, we reduce the expression to its simplest form.
Mastering factoring is vital since it is widely used in solving quadratic equations, integrating functions, and simplifying algebraic fractions.
numerator and denominator
Understanding the roles of the numerator and denominator is essential when working with fractions. The numerator is the top part of a fraction, and the denominator is the bottom part.
In the expression \(\frac{3x - 12}{15x}\), \(3x - 12\) is the numerator, and \(15x\) is the denominator. Simplifying algebraic fractions often involves factoring both the numerator and the denominator to find and remove common factors.
In our solution, after factoring the numerator, we rewrite the expression as \(\frac{3(x - 4)}{15x}\). By identifying the common factor of 3 in both the numerator and the denominator, we can cancel out this common factor:
\(\frac{3(x-4)}{15x} = \frac{x-4}{5x}\).
This cancellation simplifies the fraction, making it more manageable and easier to work with in further calculations.
In the expression \(\frac{3x - 12}{15x}\), \(3x - 12\) is the numerator, and \(15x\) is the denominator. Simplifying algebraic fractions often involves factoring both the numerator and the denominator to find and remove common factors.
In our solution, after factoring the numerator, we rewrite the expression as \(\frac{3(x - 4)}{15x}\). By identifying the common factor of 3 in both the numerator and the denominator, we can cancel out this common factor:
\(\frac{3(x-4)}{15x} = \frac{x-4}{5x}\).
This cancellation simplifies the fraction, making it more manageable and easier to work with in further calculations.
Other exercises in this chapter
Problem 12
For exercises \(5-48\), simplify. $$ \frac{6 w}{w+2}+\frac{w}{w+2} $$
View solution Problem 12
For exercises 7-32, simplify. $$ \left(\frac{y^{2}+8 y}{y^{2}}\right)\left(\frac{9 y}{y+8}\right) $$
View solution Problem 13
The relationship of \(x\) and \(y\) is an inverse variation. When \(x=2, y=10\). a. Find the constant of proportionality, \(k\). b. Write an equation that repre
View solution Problem 13
For exercises 13-24, rewrite each expression as an equivalent expression with the given denominator. $$ \frac{3}{7} ; 56 $$
View solution